Wealth distribution in presence of debts. A Fokker--Planck description

TL;DR

This paper models wealth distribution including debts using a Fokker-Planck equation with variable diffusion, showing debts are absorbed over time and a positive wealth equilibrium is reached.

Contribution

It introduces a Fokker-Planck model allowing negative wealth to represent debts, demonstrating convergence to a positive wealth equilibrium.

Findings

Debts are absorbed over time in the model.

The system reaches a unique positive wealth equilibrium.

Initial positive mean wealth leads to positive equilibrium.

Abstract

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.